natural neighbour galerkin methods

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2001; 50:1{27

Natural neighbour Galerkin methods

N. Sukumar;y;z, B. Moranx, A. Yu Semenov{ and V. V. Belikovk

Department of Civil Engineering; 2145 Sheridan Road; Northwestern University; Evanston; IL 60208-3109; U.S.A.

SUMMARY

Natural neighbour co-ordinates (Sibson co-ordinates) is a well-known interpolation scheme for multivariatedata tting and smoothing. The numerical implementation of natural neighbour co-ordinates in a Galerkinmethod is known as the natural element method (NEM). In the natural element method, natural neighbourco-ordinates are used to construct the trial and test functions. Recent studies on NEM have shown that naturalneighbour co-ordinates, which are based on the Voronoi tessellation of a set of nodes, are an appealingchoice to construct meshless interpolants for the solution of partial dierential equations. In Belikov et al.(Computational Mathematics and Mathematical Physics 1997; 37(1):9{15), a new interpolation scheme(non-Sibsonian interpolation) based on natural neighbours was proposed. In the present paper, the non-Sibsonian interpolation scheme is reviewed and its performance in a Galerkin method for the solution ofelliptic partial dierential equations that arise in linear elasticity is studied. A methodology to couple niteelements to NEM is also described. Two signicant advantages of the non-Sibson interpolant over the Sibsoninterpolant are revealed and numerically veried: the computational eciency of the non-Sibson algorithm in2-dimensions, which is expected to carry over to 3-dimensions, and the ability to exactly impose essentialboundary conditions on the boundaries of convex and non-convex domains. Copyright ? 2001 John Wiley& Sons, Ltd.

KEY WORDS: natural neighbour co-ordinates, non-Sibsonian interpolation, natural element method, mesh-less Galerkin methods, essential boundary conditions

1. INTRODUCTION

The natural element method (NEM) [1] is a Galerkin method for the solution of partial dierentialequations. In NEM, the test and trial functions are constructed using natural neighbour (Sibson)co-ordinates [2]. Natural neighbour co-ordinates are based on well-known geometric concepts suchas the Voronoi diagram and the Delaunay tessellation. The Voronoi diagram and its dual Delaunay

Correspondence to: N. Sukumar, Department of Civil Engineering, Northwestern University, 2145 Sheridan Road,Evanston, IL 60208-3109, U.S.A.

y E-mail: [email protected] Post-Doctoral Research Fellow, Theoretical and Applied MechanicsxAssociate Professor of Civil Engineering{General Physics Institute, Russian Academy of Sciences, MoscowkComputing Centre, Russian Academy of Sciences, Moscow

Contract=grant sponsor: Natural Science Foundation; contract=grant number: CMS-9732319

Received 23 November 1999Copyright ? 2001 John Wiley & Sons, Ltd. Revised 6 March 2000

2 N. SUKUMAR ET AL.

tessellation are the most fundamental and useful constructs to dene an irregular set of nodes.Recent studies using the natural element method have demonstrated its promise for the solution ofpartial dierential equations that arise in two-dimensional elastostatics [3{5] and elastodynamics[6]. In a recent study [7], a new interpolant (non-Sibsonian interpolant) based on natural neighbourswas proposed. In this paper, we present the new implementation of NEM using the non-Sibsonianinterpolant.In NEM [4] as well as other meshless methods [8], the test and trial functions in the Galerkin

implementation are constructed on the basis of a set of scattered nodes in Rd. In these methods, thenumerical integration of the weak form is carried out using a background cell=element structure.In several meshless methods, moving least squares approximants [9] are used to construct the trialand test spaces. The properties of interpolation of nodal data, ease of imposing essential boundaryconditions, and neighbour relationships that are based on the local distribution and density of nodesat a given point are some of the most important advantages of natural neighbour interpolants overmoving least squares approximants.In Belikov et al. [7], the non-uniqueness of interpolation schemes based on natural neigh-

bours was shown, and the non-Sibsonian interpolant was proposed. The local harmonic propertyand construction of higher-order interpolation schemes using the non-Sibsonian interpolant wereintroduced in Belikov and Semenov [10]. As opposed to the Sibson interpolant which is based onthe area (volume) of overlap of rst-order Voronoi polygons (polyhedra) in R2 (R3), the non-Sibsonian interpolant requires computation of Lebesgue measures of order d1 in Rd. Since mostof the properties are common to both schemes, the better computational performance and ease ofimplementation of the non-Sibsonian interpolation method over the Sibson interpolant renders thenon-Sibsonian interpolant an interesting and computationally attractive choice for the numericalsolution of partial dierential equations.The outline of this papers is as follows. In the following section, the notion of natural neigh-

bours is described and the Sibson and non-Sibsonian interpolation methods are introduced. Thecomputational algorithm used in the two methods is also presented. In Section 3, issues pertainingto the imposition of essential boundary conditions using the two interpolants are discussed, anda methodology to couple the natural element method to nite elements is described in Section 5.In Section 6, the governing equations of elastostatics together with the Galerkin formulation forNEM are described. Numerical experiment results in two-dimensional elasticity are presented inSection 7. We close in Section 8 with some concluding remarks on the performance and accuracyof the natural element method.

2. NATURAL NEIGHBOURS

The notions of natural neighbours and natural neighbour interpolation were introduced by Sibson[2] as a means for data tting and smoothing. The Voronoi diagram and its dual Delaunay triangu-lation, which are used in the construction of the natural neighbour interpolant, are useful geometricconstructs that dene an irregular set of points (nodes). For simplicity of exposition, we considertwo-dimensional Euclidean space R2; the theory, however, is applicable in a general d-dimensionalframework. Given a distribution of points (nodes) in the plane, the Delaunay simplices partitionthe convex hull of the points into regions i such that =

Sti=1 i. In Delaunay interpolation

(constant strain nite elements), a linear interpolant is constructed over each Delaunay triangle.The Delaunay triangulation of a set of nodes is non-unique, and hence the interpolation is sensitive

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2001; 50:1{27

NATURAL NEIGHBOUR GALERKIN METHODS 3

Figure 1. Voronoi diagram V (N ) for a set N of seven nodes.

to geometric perturbations of the position of the nodes. As opposed to the Delaunay triangulation,its dual the Voronoi diagram is unique. We now formally introduce the Voronoi diagram andthereafter the denition of natural neighbours.Consider a set of distinct nodes N= fn1; n2; : : : ; nMg in R2. The Voronoi diagram (or 1st-order

Voronoi diagram) of the set N is a subdivision of the plane into regions TI (closed and convex,or unbounded), where each region TI is associated with a node nI , such that any point in TI iscloser to nI (nearest neighbour) than to any other node nJ 2N (J 6= I)TI is the locus of pointscloser to nI than to any other node. The regions TI are the Voronoi cells of nI . In mathematicalterms, the Voronoi polygon TI is dened as [11]

TI = fx2R2: d(x; xI )1) Voronoi diagrams in the plane. Of particular interestin the present context is the case k =2, which is the second-order Voronoi diagram. The second-order Voronoi diagram of the set of nodes N is a subdivision of the plane into cells TIJ , whereeach region TIJ is associated with a nodal-neighbour-pair (nI ; nJ ) (k-tuple for the k-order Voronoidiagram), such that TIJ is the locus of all points that have nI as the nearest neighbour, and nJ asthe second nearest neighbour. It is emphasized that the cell TIJ is non-empty if and only if nI andnJ are neighbours. The second-order Voronoi cell TIJ (I 6= J ) is dened as [2].

TIJ = fx2R2: d(x; xI )

4 N. SUKUMAR ET AL.

Figure 2. Construction of natural neighbours: (a) Original Voronoi diagram and x; and(b) 1st-order and 2nd-order Voronoi cells about x.

that if x lies within the circumcircle of triangle DT(nI ; nJ ; nK), then nI ; nJ , and nK are naturalneighbours of x. In Figure 2(b), the perpendicular bisectors from point x to its natural neighboursare constructed and the Voronoi cell Tx (closed polygon abcd) is obtained. It is observed that xhas four (n=4) natural neighbours, namely nodes 1, 2, 3, and 4.

2.1. Sibson interpolation

Natural neighbour co-ordinates are used as the interpolating functions in natural neighbour (Sibson)interpolation. We refer to Figure 2 in order to dene the natural neighbour co-ordinates for apoint x in the plane. Let (x) be a Lebesgue measure (length, area, or volume in 1D, 2D, or3D, respectively) of Tx, and I (x) (I =1{4) be that of TxI . In two-dimensions, the measures areareas, and hence we denote A(x) (x) and AI (x) I (x). The natural neighbour co-ordinatesof x with respect to a natural neighbour I is dened as the ratio of the area of overlap of theVoronoi cells TI and Tx to the total area of the Voronoi cell of x:

I (x)=AI (x)A(x)

(3)

where I ranges from 1 to n, and A(x)=Pn

J=1 AJ (x). The four regions shown in Figure 2(b) arethe second-order cells, whereas their union (closed polygon abcd) is a rst-order Voronoi cell.Referring to Figure 2, the shape function 3(x) is given by

3(x)=A3(x)A(x)

(4)

If the point x coincides with a node (x= xI ); I (x)= 1, and all other shape functions are zero.The properties of positivity, interpolation, and partition of unity directly follow:

06I (x)61; I (xJ )= IJ ;nP

I=1I (x)= 1 in (5)

Natural neighbour shape functions also satisfy the local co-ordinate property [2], namely

x=nP

I=1I (x)xI (6)



which, in conjunction with Equation (5) imply that the natural neighbour interpolant spans thespace of linear polynomials (linear completeness).Natural neighbour interpolation has primarily been used in the area of data interpolation and

modelling of geophysical phenomena [13{15]. The support of the shape function I (x) is theintersection of the convex hull with the union of all Delaunay circumcircles that pass through nodeI [16]. Natural neighbour shape functions are C everywhere, except at the nodes where theyare C0 [2; 16]. Farin [16] proposed a C1 natural neighbour interpolant based on Bernstein{Beziersimplices, and Sukumar and Moran [5] developed a computational methodology for its applicationto fourth-order elliptic PDEs. In one-dimension, natural neighbour interpolation is identical tolinear nite elements [4]; in the particular case of three natural neighbours, n-n interpolation isprecisely barycentric co-ordinates; and for four natural neighbours at the vertices of a rectangle,bilinear interpolation is realized [16]. A detailed description and discussion of the above propertiesof natural neighbour interpolants can be found in Sukumar et al. [4].

2.2. Non-Sibsonian interpolation

We give the following rigorous denition of the non-Sibsonian interpolant [7]. Let N= fn1; n2;: : : ; nMg be a set of distinct nodes in Rd. We denote the Voronoi cell of node xI by TI :TI = fx2Rd: d(x; xI )

6 N. SUKUMAR ET AL.

Figure 3. Non-Sibsonian interpolation. Figure 4. Bilinear interpolation on aregular grid (n=4).

Consider an interpolation scheme for a vector-valued function u(x) : !R2, in the form:

uh(x)=nP

I=1I (x)uI (10)

where uI (I =1; 2; : : : ; n) are the vectors of nodal displacements at the n natural neighbours,and I (x) are the Sibson (see Equation (3)) or the non-Sibsonian shape functions dened inEquation (9). In the natural element method, the trial and test functions are constructed using theapproximation indicated in Equation (10).

2.3. Properties

The non-Sibsonian interpolant is based on the notion of natural neighbours, and hence most of theproperties of the natural neighbour interpolant are also shared by the non-Sibsonian interpolant.Properties such as partition of unity, positivity, interpolation, and support and regularity of theshape functions are common to both interpolants. In addition, due to the above properties, bothinterpolants ensure the boundedness of the norm of the interpolated result: kuk6max uI . Inter-polants that are linear combinations of the Sibson and non-Sibsonian interpolants are also validnatural neighbour-based interpolants that can be used for the solution of PDEs. We address thelinear completeness of the non-Sibsonian interpolant and discuss univariate, bivariate, and higher-order interpolation within the context of non-Sibsonian interpolation.

2.3.1. Linear completeness. The non-Sibsonian interpolant has linear precision, i.e. it spans thespace of linear polynomials in Rd [7]. In the nite element literature, the ability of the inter-polant to reproduce constant and linear displacement elds is known as linear completenessanecessary condition for convergence for second-order elliptic PDEs such as Poissons equation andelastostatics. The proof that the non-Sibsonian interpolation has linear completeness follows:

Proof. Consider a linear displacement eld in the form:

u(x)= Q+ RTx (11)



where Q and R are constant vectors. The exact nodal displacements are given by

uI = Q+ RTxI (12)

where I is the index for any particular node. Consider the NEM trial function, namely

uh(x)=nP

I=1I (x)uI (13)

where uI is the vector of nodal displacements for node I . On using Equation (12) in the aboveequation, we obtain

uh(x)= QnP

I=1I (x) + RT

nPI=1

I (x)xI (14)

By adding and subtracting RTx to the above equation and noting thatP

I I (x)= 1, we have

uh(x)= Q+ RTx + RTnP

I=1I (x)(xI x) (15)

and hence to complete the proof it suces to show that the following equality holds:

nPI=1

I (x)(xI x)= 0 (16)

We rst show that the above equality holds in the planar (two-dimensional) case. To this end,transforming from (x; y) in R2 to z= x + iy in the complex plane C, the above criterion can bere-written as

nPI=1

I (z)(zI z)= 0 (17)

which on substituting the explicit form of the non-Sibsonian shape function from Equation (9)becomes

PnI=1(zI z)

sI (z)hI (z)Pn

J=1sJ (z)hJ (z)

= 0 (18)

To prove the above, we note that the sides of the Dirichlet cell that contain (x; y) are the complexnumbers zI . Since the polygon has a closed contour, the following identity holds:

nPI=1

zI =0 (19)


8 N. SUKUMAR ET AL.

Introducing the trigonometric form zI = sI (z) exp(iI (z)), we rearrange the above equation asfollows:

nPI=1

zI =nP

I=1sI (z) exp(iI (z))= i

nPI=1

sI (z) exp(iI (z) i=2)=0 (20)

and since jzI zj=2hI (z), we can write

inP

I=1[hI (z) exp(iI (z) i=2)] sI (z)hI (z) =

i2

nPI=1(zI z) sI (z)hI (z) = 0 (21)

which proves Equation (18) and hence

uh(x)= Q+ RTx= u(x) (22)

which completes the proof for the planar case.Now, we prove the same for the case of arbitrary Rd. By analogy we consider a linear dis-

placement eld u(x) in Rd and show that Equation (16) holds for it identically. Following the2-dimensional case, the criterion to be proved for the d-dimensional case has the following vectorform: Pn

I=1 (xI x)sI (x)hI (x)Pn

J=1sJ (x)hJ (x)

= 0 (23)

To prove the above, consider the closed surface S of the Voronoi polyhedron that encloses x2Rdand has the volume V. Now, consider the vector identity (Greens theorem):Z

Vrf dV =

ISf dS (24)

On substituting f=1 in the above equation, we obtainZVrf dV = 0=

ISdS=

nPI=1

(xI x)sI (x)jxI xj (25)

and since jxI xj=2hI (x), we havenP

I=1

(xI x)sI (x)2hI (x)

= 0 (26)

which proves Equation (23) and hence

uh(x)= Q+ RTx= u(x) (27)

which completes the proof for the general d-dimensional case.

2.3.2. Univariate interpolation. In 1-dimension, the non-Sibsonian shape function given inEquation (8) is undened since the Lebesgue measure of a point is zero. However, by con-sidering univariate interpolation as a limiting case of bivariate interpolation, it is readily seen thatlinear interpolation is realized in 1-dimension (see Section 3.1 too). Hence the equivalence withone-dimensional linear nite elements is obtained, as was the case for the Sibson interpolant [4].



2.3.3. Bivariate interpolation. Case I: n=3 or n=4. If a point x has three neighbours, thenby a unique argument or by considering the linear reproducing conditions given in Section 2.3.1,barycentric co-ordinates are realized by nS interpolation. The proof is similar to that outlined inSukumar et al. [4] for the Sibson interpolant. For the case of four natural neighbours (n=4) atthe vertices of a rectangle, bilinear interpolation on the rectangle is obtained. The proof follows:

Proof. Consider a point x with four natural neighbours located at the vertices of a unit square:(x1; y1)= (0; 0), (x2; y2)= (1; 0), (x3; y3)= (1; 1), and (x4; y4)= (0; 1) (Figure 4). The rst-order(dark line) Voronoi cell of point x is shown in Figure 4. By denition of the non-Sibsonian shapefunctions, we can write

I (x)=

sI (x)hI (x)P4

J=1sJ (x)hJ (x)

(I =14) (28)

where sI (x) and hI (x) are indicated in Figure 4.By recalling the denition of 1st-order Voronoi cells, it is clearly seen that vertex a is the centre

of the circle that circumscribes the triangle 412x, and proceeding likewise, b is the circumcentreof triangle 423x, c that of 434x, and d that of 441x. These co-ordinates are computed to be

a1 =12; a2 =

x + x2 + y22y

(29a)

b1 =1 + y x2 y2

2(1 x) ; b2 =12

(29b)

c1 =12; c2 =

1 + x x2 y22(1 y) (29c)

d1 =y + x2 + y2

2x; d2 =

12

(29d)

We note that s1(x)= j~abj, s2(x)= j~bcj, s3(x)= j~cdj, s4(x)= j~daj, and 2hI =d(x; xI ). By carryingout simple algebraic calculations, we obtain

s1(x)h1(x)

=x + y x2 y2

xy(30a)

s2(x)h2(x)

=x + y x2 y2

y(1 x) (30b)

s3(x)h3(x)

=x + y x2 y2(1 x)(1 y) (30c)

s4(x)h4(x)

=x + y x2 y2

x(1 y) (30d)


10 N. SUKUMAR ET AL.

and hence the non-Sibsonian shape functions are

1(x) = (1 x)(1 y) (31a)

2(x) = x(1 y) (31b)

3(x) = xy (31c)

4(x) = y(1 x) (31d)

which are bilinear nite element shape functions. The above derivation is easily generalized to therectangle (linear transformation of a square), and hence bilinear interpolation on the rectangle isrealized by the non-Sibsonian interpolant.

Case II: Arbitrary n. We consider the computation of non-Sibsonian shape functions for thecase in which the point x=(x; y) has an arbitrary number of natural neighbours. Let xm=(xm; ym)and xn=(xn; yn) be two adjacent natural neighbours for the point x, where n=m+1 or n=m1(Figure 5). We assume clockwise orientation to be the positive sense. In Figure 5(a), line AB isperpendicular to xm x and line CB is perpendicular to xn x. Any point along the line AB hasthe form:

AB : 12 (xm + x) + tm(xm x) (32)

where tm is a scalar parameter. In addition,

xm + x= (xm + x; ym + y) (33a)

(xm x) = (xm x; ym y)=(ym y;xm + x) (33b)

where q= q ^ e3. Now, any point along CB has the form:BC : 12 (xn + x) tn(xn x) (34)

where tn is a scalar parameter. In addition,

xn + x= (xn + x; yn + y) (35a)

(xn x) = (xn x; yn y)=(yn y;xn + x) (35b)

Now, the condition for line intersection at point B can be stated as

12 (xm + x) + tm(xm x)= 12(xn + x) tn(xn x) (36)



Figure 5. Computation of non-Sibsonian interpolant in R2: (a) Geometric construction;(b) rm>0 and lm0 and lm>0.

and hence we obtain the following system of equations for tm and tn:

(ym y)tm (yn y)tn = xm xn2 (37a)

(xm x)tm + (xn x)tn = ym yn2 (37b)

Solving the above system, we obtain

tm =12(xm xn)(xn x) + (ym yn)(yn y)(xm x)(yn y) (xn x)(ym y) (38a)

tn =12(xm xn)(xm x) + (ym yn)(ym y)(xm x)(yn y) (xn x)(ym y) (38b)

Now, we note that

jAB j

12 jxm xj

=2jtmj (39)



which can be readily inferred from Figure 5(a). By taking into account n=m+ 1 and n=m 1(see the cases shown in Figures 5(b) and 5(c)), we can compute sm=hm as

sm=hm= jrm lmj (40a)where

rm =(xm xm+1)(xm+1 x) + (ym ym+1)(ym+1 y)(xm x)(ym+1 y) (xm+1 x)(ym y) (40b)

lm =(xm xm1)(xm1 x) + (ym ym1)(ym1 y)

(xm x)(ym1 y) (xm1 x)(ym y) (40c)

On using the above relations for all the natural neighbours, the non-Sibsonian shape functions areevaluated using Equation (9).

2.3.4. Higher-order interpolation. Belikov and Semenov [17] proposed a compact procedure forthe generation of higher-order interpolation using non-Sibsonian interpolants. For kth-order inter-polation at the point x, it is sucient that u(x) satises the equation 4ku=0, where 4 is theLaplacian operator. Then, for the neighbours of x, a sequence of values w1; w2; : : : ; wm1 are com-puted where the wi satisfy the relations 4u=w1, 4w1 =w2, 4w2 =w3; : : : ;4wk1 =wk =0. Thecomputation of u(x) is performed by going through the above sequence of equations in reverseorder.Let us rst consider an interpretation of non-Sibsonian interpolation. Using Equation (10), we

can write the non-Sibsonian trial function uh(x) for a scalar-valued function u(x) in the form:

nPI=1

uI uh(x)2hI (x)

sI (x)= 0 (41)

Thus, the continuous analogue of the above discrete form isIS

@u@ndS =

ZV4u dV =0 (42)

or

4u=0; 4= @2

@x21+ + @

2

@x2d(43)

where Gausss (divergence) theorem has been used to convert the surface integral into a volumeintegral. Thus, Equation (10) provides an approximate local discrete solution of the harmonicequation in Rd. We can refer to I (x) as harmonic co-ordinates [17].Now, for the specic case of k =2 (second-order interpolation) we obtain two equations:

4u=w; 4w=0, and hence the nal formulas for the calculation of the second-order interpolanthave the following form:

wI =1VI

pIPJ=1

uJ uIhIJ

s IJ (44a)



wh(x) =nP

I=1I (x)wI (44b)

uh(x) =nP

I=1I (x)uI w

h(x)V (x)PnJ=1

sJ (x)hJ (x)

(44c)

Here, n is the number of natural neighbours for point x; pI is the number of natural neighboursfor point xI , and VI is the area (volume) of the corresponding Dirichlet cell. In determining thenatural neighbours for point xI , one considers the nodal set N nI (see Section 2). Therefore,the natural neighbours pI for point xI are easily seen to be those nodes which are connected tonode xI in the Delaunay triangulation of the set N. First the parameters wI are evaluated usingEquation (44a), and then wh(x) is obtained from Equation (44b) which is used in Equation (44c)to compute uh(x) at x.For the derivation of Equation (44), the following equalities in three dierent forms are used:

Continuous form: 4u=w (45a)

Integral form:IS

@u@ndS =

ZVw dV (45b)

Discrete form:pP

J=1

uJ uKhKJ

sKJ =wKVK (45c)

where p is the number of natural neighbour for the point xK . Comparing the above to Equa-tion (44), we note that if xK = x, then p= n; VK =V (x), and sKJ = sJ (x) and h

KJ = hJ (x); if

xK = xI , then p=pI .We now proceed to re-cast the second-order non-Sibsonian interpolant in the standard trial

function form:

uh(x)=mP

I=1 I (x)uI (46)

where I (x) is the second-order non-Sibsonian shape function for node I at x and uI (I =1; 2;: : : ; m) are the vectors of nodal displacements. Using Equation (44), we can immediately write:

uh(x)=nP

I=1I (x)[uI (x)wI ]; (x)= V (x)Pn

J=1sJ (x)hJ (x)

(47)

Let Nx= fnx1; nx2; : : : ; nxng be a set of n natural neighbours for the point x and NI = fnI1; nI2; : : : ;nIpI g be the pI natural neighbours for the point xI (nxI =2NI ). The set N of all nodes in theapproximation is: N=Nx [

SnI=1NI . The cardinality of N is m. The expression for wI is given

in Equation (44a). We note that for any given node in Nx, there are two distinct sources ofcontribution from the second term of the above equation. By some simple algebraic manipulations,



we arrive at the following:

I (x) =I (x)1 + (x)

1VI

pIPJ=1

s IJh IJ

(x)pIPJ=1

1VJ

sJIhJI

; nI 2Nx (48a)

I (x) =(x)nP

J=1nI2NJ

J (x)VJ

sJIhJI

; nI =2Nx (48b)

and since s JI = sIJ and h

JI = h

IJ , we obtain the result

uh(x)=mP

I=1 I (x)uI (49a)

where

I (x) =I (x)1 + (x)

pIPJ=1

s IJh IJ

1VI 1

VJ

; nI 2Nx (49b)

I (x) =(x)nP

J=1nI2NJ

J (x)VJ

sJIhJI

; nI =2Nx (49c)

The above higher-order scheme is simple and computationally attractive in numerical calculations[17; 18].

2.4. Computational algorithm

In Sukumar et al. [4], Watsons algorithm [19] was used for the computation of the Sibsoninterpolant. A drawback of Watsons algorithm is that it fails for points that lie on the edgeof a Delaunay triangle. In order to overcome this shortcoming, we adopted the Bowyer{Watsonalgorithm [20; 21] as the basis for the evaluation of the Sibson and non-Sibsonian interpolants.Natural neighbour (Sibson) interpolation is based on the area (volume) of intersection of poly-

gons (polyhedrons) in R2 (R3). The polyhedron intersection problem is a non-trivial task in com-putational geometry, and hence easy-to-implement algorithms that are computationally attractivefor Sibson computations are still unavailable. Owens [22] proposed a three-dimensional algorithmfor natural neighbour interpolation and Braun and Sambridge [1] used Lasserres algorithm [23]to compute the natural neighbour shape function in their PDE application. See Aftosmis [24] fora description of the many issues involved in the polyhedron intersection problem. Since the non-Sibsonian interpolant relies on the evaluation of a Lebesgue measure of one dimension less than theSibson interpolant, the implementation of the non-Sibsonian interpolant in, both, 2-dimensions and3-dimensions is viable. The essential ingredients of the computational algorithm for non-Sibsonianinterpolation are presented in Table I. Standard template library (STL) containers in C++ areused in the implementation. The STL containers map and multi-map are used, where map is acontainer that maps an integer key to another integer or oating-point number, and multi-map is amap from an integer key to many integers or oating-point numbers. These containers provide fastsorting and searching on keys, which eases the implementation and leads to better computationaleciency.



Table I. Pseudo-code for computation of non-Sibsonian interpolant.

Compute natural neighbour set N and set T of deleted simplices for point x Find simplex t containing x and set T t Test all neighbouring simplices ti of t. If kxvik2


Figure 6. Sibson and non-Sibsonian shape functions: (a) Nodal grid; (b) Sibson shapefunction SA(x); and (c) non-Sibson shape function

nSA (x).

trial functions uh(x) are strictly linear between two nodes that belong to an edge of a Delaunaytriangle. The proof follows:

Proof. Consider a typical Delaunay triangle which has one edge (two nodes) along the boundaryof the convex hull, and the trial functions uh() are to be evaluated at a point along the edge1{2 (Figure 7). For simplicity and for ease of illustration, we assume that has only three naturalneighbours, namely nodes 1, 2, and 3. We use a local co-ordinate system along the edge 1{2such that =0 at node 1 and =1 at node 2. The rst-order Voronoi cell for is shown inFigure 7. By denition, the non-Sibsonian shape functions can be written as

I ()=

sI ()hI ()3P

J=1

sJ ()hJ ()

; (I =13) (50)

Since the length of the Voronoi edge associated with nodes 1 and 2 on the boundary of theconvex hull is unbounded, we can express the Lebesgue measures s1(); s2(), and s3()



Figure 7. Linear behaviour of uh() along the boundary of a convex domain.

as

s1()= limL

L+ 1(); s2()= limL

L+ 2(); s3()= 3() (51)

where 1(), 2(), and 3() are nite. In addition, we also have h1()= =2, h2()= (1 )=2,and h3() is nite. On using Equation (50), we can write

1() = limL

(L+ 1())(1 )h3()(L+ 1())(1 )h3() + (L+ 2())h3() + 3()(1 ) (52a)

2() = limL

(L+ 2())h3()(L+ 1())(1 )h3() + (L+ 2())h3() + 3()(1 ) (52b)

3() = limL

3()(1 )(L+ 1())(1 )h3() + (L+ 2())h3() + 3()(1 ) (52c)

Taking the limit as L!1 in the above equations, we obtain1()= 1 ; 2()= ; 3()= 0 (53)

and hence along the edge 1{2, the shape function contributions from only nodes 1 and 2 are non-zero. The above result is in general true, even if more than three natural neighbours are considered.This is so, since the Lebesgue measure associated with all interior nodes is nitesimilar to 3()in Figure 7. Using the above equation, the trial functions at the point can be written as

uh()= (1 )u1 + u2 (54)which are linear functions, and hence the proof.

3.2. Non-convex domain

Consider a set of nodes N that describes a non-convex domain R2. Consider a 1 (1= @)which renders the domain to be non-convex. For purpose of illustration, we choose a non-convexdomain bounded by two concentric circles. The discrete model (one-quarter) along with theVoronoi diagram are shown in Figure 8. It is evident that the Voronoi cells for the nodes along i



Figure 8. Linear behaviour of uh(x) along the boundary of a non-convex domain:(a) Nodal discretization; and (b) Voronoi diagram.

(i=2{4) are unbounded, whereas the Voronoi cells for the nodes along 1 are bounded and there-fore have nite areas (Figure 8(b)). The Sibson interpolant is strictly linear along =2 [3 [4;however, the interpolant is not linear between adjacent nodes on the boundary 1 since for x21,there exists non-zero shape function contributions at x from some interior nodes [4].Let the essential boundary u=1. In Figure 9, we show four contiguous nodes (nodes 1, 2, a,

and b) along u. Let be a local co-ordinate system along the edge 1{2 such that =0 at node 1and =1 at node 2. We consider a point 2u which is located along the Delaunay edge 1{2. If apoint x lies inside the circumcircle of a Delaunay triangle, then the Delaunay triangle is known asthe circum-triangle of point x. By the Delaunay empty circumcircle criterion, all circum-trianglesof point must have nodes 1 and 2 as its vertices. This immediately precludes the nodes alongu that are shown by open circles (Figure 9) to be natural neighbours of . We assume that hasfour natural neighbours, namely nodes numbered 1{4 and shown by the dark circles in Figure 9.The Voronoi cell for point and the length measures sI (x) and hI (x) that appear in the denitionof the non-Sibsonian interpolant are shown in Figure 9. We note the Lebesgue measure sI ()associated with nodes 1 and 2 are unbounded, and those for the interior nodes 3 and 4 are nite.It is apparent that the computation of the shape functions is now similar to the convex case, andby following similar arguments (see the proof in Section 3.1), we again arrive at the conclusionthat the NEM trial functions using non-Sibsonian shape functions are precisely linear along theedge 1{2, i.e.

uh()= (1 )u1 + u2 (55)

A consequence of the above discussion is that by choosing non-Sibson shape function as trialand test functions in the natural element method, essential boundary conditions can be directlyimposed on the nodes, as in the nite element methodthis is due to the interpolating propertyof the shape functions and the linearity of the approximation along the essential boundary ofthe domain. It is to be noted that the above inference is rigorously true for convex as well asnon-convex domains. The natural element method with non-Sibsonian shape functions is the onlymeshless Galerkin method that we are aware of that will exactly satisfy (linear) essential boundaryconditions.



Figure 9. Non-Sibsonian interpolation along theboundary of a non-convex domain.

Figure 10. Finite element and natural elementmethod coupling.

4. FE AND NEM COUPLING

The coupling between linear nite elements and the natural element method using the non-Sibsonian interpolant, is straight-forward. In Figure 10, a domain =FE [NEM is illustrated,where FE is the nite element domain and NEM is the NEM domain. Since the non-Sibsonianinterpolant is precisely linear on the boundary of convex as well as non-convex domains, a seam-less procedure emerges to implement the coupling. By adopting regional interpolation, i.e. niteelement interpolation carried out in FE and NEM interpolation in NEM, the trial function in thedomain is given by

uh(x)=

8R(67a)

u = 0 (67b)

where

C1 =( + ) + +

(68)

In the above equation, and are the Lame constants of the respective phases, and the eigen-strain is a constant dilatational strain. The material properties used in the numerical computationare [29]: =497:16, =390:63 in the -phase, and the constants in the -phase are =656:79,=338:35. These correspond to E=1000, =0:28, E=900, and =0:33. A constant di-latational eigenstrain =0:01 is assumed in the analysis, and the associated eigenstrain tensor isU = (e1e1 + e2e2).The numerical model (quarter symmetry) is shown in Figure 13, where the nodal discretization

consists of 647 nodes, with 114 nodes in the inclusion, 520 nodes in the matrix, and 13 nodesalong the interface r=R. The outer radius R0 = 200 is suciently large in comparison to theradius of the inclusion R=5, so as to adequately represent the innite matrix. Essential boundary



Figure 11. Nodal grids for eigenanalysis: (a) uniform grid (25 nodes); (b) randomset (70 nodes); and (c) irregular focused grid (278 nodes).

Figure 12. Inclusion embedded in an innite matrix.

conditions are imposed along the lines of symmetry, and the outer radius R0 = 200 is traction free.Plane strain conditions are assumed in the numerical computations. The numerical computationsare carried out using the Sibson interpolant, non-Sibsonian interpolant, and the FE{NEM couplingprocedure outlined in Section 4. The NEM solution recovered the cylindrical symmetry in the



Figure 13. Nodal discretization for inclusion in an innite matrixproblem: (a) quarter model; (b) inclusion ( phase).

solution, and hence results are presented as a function of only the radial distance. The numericalresults using the Sibson interpolant, non-Sibsonian interpolant, and the FE{NEM coupling techniquecompare favourably to each other. In Figure 14, a comparison of the numerical and exact solutionis presented. The results shown in Figure 14 are computed along a radial line (r=0 to r=25)at =30. Along the radial line, 30 equi-spaced output points are chosen within the inclusion,and 30 equi-spaced points in the matrix. Excellent agreement between the NEM and the analyticalsolution is observed. The strains as well as the stresses are in good agreement with the exactsolution. The oscillations in the radial and hoop strains are negligible; they are, however, a bitmore pronounced in the stress solutions.In order to show the linear behaviour of the non-Sibsonian interpolant along the boundary ,

we consider two nodes A and B along (Figure 15(a)). In Figure 15(b), the non-Sibsonianshape functions A and B are plotted along the element edge, with a local co-ordinate alongA{B. The numerical results are computed along the line xA to xB, where xA=(xA + ; yA) with=1012 being a small tolerance. The tolerance is required in the numerical computations sincefor a point x 2 , the length of the Voronoi edges associated with nodes A and B is unbounded.It is seen from Figure 15 that a linear displacement approximation along A{B is realized in thenumerical computations, which supports the theoretical proof presented in Section 3.2. This showsthat essential boundary conditions in NEM using the non-Sibsonian interpolant can be prescribedexactly as in nite elements. In addition, these numerical results validate the FE{NEM couplingtechnique that is described in Section 4.

7. CONCLUSIONS

In this paper, we introduced the use of natural neighbour-based interpolants (Sibson and non-Sibsonian) for the solution of elliptic partial dierential equations. The numerical implementationof these interpolants in a Galerkin method is known as the natural element method (NEM). Thenotion of natural neighbours relies on the Voronoi diagram which is unique for a given set ofscattered nodes in Rd. The local density and spatial location of nodes is taken into account in theconstruction of the natural neighbour-based interpolants. Some of the most important properties



Figure 14. Comparison of NEM and the exact solution for an inclusion with a dilatational eigenstrainin an innite matrix: (a) radial displacement ur(r); (b) radial strain rr(r); (c) hoop strain (r);

(d) radial stress rr(r); and (e) hoop stress (r).

of the non-Sibsonian interpolant were reviewed and new results on the imposition of essentialboundary conditions were presented.An eigenanalysis of the Sibson and non-Sibsonian interpolants was carried out, and the meshless

interpolating spaces were found to be uniformly linearly independent. The computational eciencyof the non-Sibsonian interpolants over the Sibson interpolant was demonstrated and a versatilealgorithm for 2- and 3-dimensional non-Sibsonian computations was presented. The NEM results



Figure 15. Linear approximation along a non-convex boundary: (a) quarter model;and (b) non-Sibsonian shape functions along .

for the problem of an inclusion with a constant eigenstrain, embedded in an innite matrix, werefound to be in good agreement with the exact solution. Essential boundary conditions in NEMusing the non-Sibsonian interpolants can be imposed as in nite elements, for both, convex, andnon-convex domains. A rigorous proof to this fact was presented, which was numerically veried.A simple technique to couple nite elements to the natural element method was also described.The robust and appealing properties of natural neighbour-based interpolants, together with theseamless means to couple NEM to FEM and the ease of modelling material interfaces opens upmany exciting areas of mechanics research that can be explored by this new technique.

ACKNOWLEDGEMENTS

The authors are grateful for the research support of the National Science Foundation through contract CMS-9732319.

REFERENCES

1. Braun J, Sambridge M. A numerical method for solving partial dierential equations on highly irregular evolving grids.Nature 1995; 376:655{660.

2. Sibson R. A vector identity for the Dirichlet tesselation. Mathematical Proceedings of the Cambridge PhilosophicalSociety 1980; 87:151{155.

3. Sukumar N. The natural element method in solid mechanics. Ph.D. Thesis, Theoretical and Applied Mechanics,Northwestern University: Evanston, IL, USA, 1998.

4. Sukumar N, Moran B, Belytschko T. The natural element method in solid mechanics. International Journal forNumerical Methods in Engineering 1998; 43(5):839{887.

5. Sukumar N, Moran B. C1 natural neighbour interpolant for partial dierential equations. Numerical Methods for PartialDierential Equations 1999; 15(4):417{447.

6. Bueche D, Sukumar N, Moran B. Dispersive properties of the natural element method. Computational Mechanics2000; 25(2/3):207{219.

7. Belikov VV, Ivanov VD, Kontorovich VK, Korytnik SA, Semenov AY. The non-Sibsonian interpolation: a new methodof interpolation of the values of a function on an arbitrary set of points. Computational Mathematics and MathematicalPhysics 1997; 37(1):9{15.

8. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless methods: an overview and recent developments.Computer Methods in Applied Mechanics and Engineering 1996; 139:3{47.



9. Lancaster P, Salkauskas K. Curve and Surface Fitting. Academic Press: London, 1986.10. Belikov VV, Semenov AY. Non-Sibsonian interpolation on arbitrary system of points in euclidean space and adaptive

generating isolines algorithm. In Numerical Grid Generation in Computational Field Simulations, Cross M, Soni BK,Thompson JF, Hauser J, Eiseman PR (eds). University of Greenwich: London, UK 1998; 277{286.

11. Green PJ, Sibson RR. Computing Dirichlet tessellations in the plane. The Computer Journal 1978; 21:168{173.12. Lawson CL. Software for C1 surface interpolation. In Mathematical Software III, Rice JR (ed.), vol. 3. Academic

Press: New York, NY, 1977.13. Watson DF. Contouring: A Guide to the Analysis and Display of Spatial Data. Pergamon Press: Oxford, 1992.14. Jones NL, Owens SJ, Perry EC. Plume characterization with natural neighbour interpolation. Proceedings

GEOENVIRONMENT 2000, Geotechnical Engineering and Environmental Engineering Divisions=ASCE. New York,NY 1995; 331{345.

15. Braun J, Sambridge M, McQueen H. Geophysical parameterization and interpolation of irregular data using naturalneighbours. Geophysical Journal International 1995; 122:837{857.

16. Farin G. Surfaces over Dirichlet tessellations. Computer Aided Geometric Design 1990; 7(1{4):281{292.17. Belikov VV, Semenov AY. New non-Sibsonian interpolation on arbitrary system of points in Euclidean space. In 15th

IMACS World Congress, Numerical Mathematics, vol. 2, Wissen Tech. Verlag: Berlin 1997; 237{242.18. Belikov VV, Semenov AY. Non-Sibsonian interpolation on arbitrary system of points in Euclidean space and adaptive

isolines generation. Applied Numerical Mathematics 2000; 32(4):371{387.19. Watson DF. nngridr: An implementation of natural neighbour interpolation. David Watson, 1994.20. Bowyer A. Computing Dirichlet tessellations. Computer Journal 1981; 24:162{166.21. Watson DF. Computing the n-Dimensional Delaunay tessellation with application to Voronoi polytopes. The Computer

Journal 1981; 24(2):167{172.22. Owens SJ. An implementation of natural neighbour interpolation in three dimensions. Masters Thesis, Brigham Young

University, 1992.23. Lasserre JB. An analytical expression and an algorithm for the volume of a convex polyhedron in Rn. Journal of

Optimization Theory and Applications 1983; 39(3):363{377.24. Aftosmis AF. Solution adaptive cartesian grid methods for aerodynamic ows with complex geometries. Lecture Notes

for 28th Computational Fluid Dynamics Lecture Series. von Karman Institute for Fluid Dynamics: Rhode-Saint-Genese,Belgium, 1997.

25. Belytschko T, Organ D, Krongauz Y. A coupled nite element-element-free Galerkin method. ComputationalMechanics 1995; 17:186{195.

26. Strang G, Fix G. An Analysis of the Finite Element Method. Prentice-Hall: Englewood Clis, NJ, 1973.27. Smith BT, Boyle JM, Garbow BS, Ikebe Y, Klema VC, Moler CB. Matrix Eigensystem Routines|EISPACK Guide.

Springer: New York, 1974.28. Mura T. Micromechanics of Defects in Solids. Martinus Nijho: The Hague, Netherlands, 1987.29. Cordes LW, Moran B. Treatment of material discontinuity in the element-free Galerkin method. Computer Methods

in Applied Mechanics and Engineering 1996; 139:75{89.


natural neighbour galerkin methods

Documents

natural neighbours

natural neighbourcoordinates

naturalneighbour coordinates

nonsibson interpolant

voronoi tessellation

nonconvex domains

voronoi diagram

recent studies